Canonical partition function for N ideal gasses

In summary, the canonical partition function is a mathematical concept used in statistical mechanics to calculate the thermodynamic properties of a system of N non-interacting particles. It is derived using the principles of statistical mechanics and the Boltzmann distribution, and it allows for the calculation of important thermodynamic quantities and predictions about a system's behavior at different temperatures. It differs from the grand canonical partition function in that it is used for systems with a fixed number of particles and cannot be applied to interacting particles.
  • #1
Schwarzschild90
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Homework Statement


Exercise 4 in the upload titled Dok1.pdf.

Write down an expression for the canonical partition function for N ideal Na2 gas molecules, when the rotational contribution is treated classically, and all inner degrees of freedom are treated quantum mechanically. Use this and earlier results to calculate an expression for the equilibrium constant for the reaction (1) and calculate its value at =1000.

Homework Equations

and 3. the attempt at a solution[/B]
A tutor came up with this nonsensical solution:
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  • #2
I've redone the exercise and come up with the following solution instead:

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1. What is the canonical partition function for N ideal gases?

The canonical partition function for N ideal gases is a mathematical concept used in statistical mechanics to calculate the thermodynamic properties of a system of N non-interacting particles. It is denoted by the symbol Z and is defined as the sum of all possible microstates of the system, each weighted by the Boltzmann factor e-Ei/kBT, where Ei is the energy of the ith microstate, kB is the Boltzmann constant, and T is the temperature of the system.

2. How is the canonical partition function derived?

The canonical partition function is derived using the principles of statistical mechanics and the Boltzmann distribution. It involves calculating the probability of a system being in a particular microstate and then summing over all possible microstates to obtain the partition function. This is then used to calculate thermodynamic quantities such as the internal energy, entropy, and free energy of the system.

3. What is the significance of the canonical partition function?

The canonical partition function is a fundamental concept in statistical mechanics and is used to describe the thermodynamic behavior of a system at the molecular level. It allows us to calculate important thermodynamic quantities and make predictions about the behavior of a system at different temperatures. It also forms the basis for understanding phase transitions and the behavior of complex systems.

4. How does the canonical partition function differ from the grand canonical partition function?

The canonical partition function is used for systems with a fixed number of particles, while the grand canonical partition function is used for systems with varying numbers of particles. In the canonical ensemble, the number of particles is fixed, while in the grand canonical ensemble, the number of particles can fluctuate. Additionally, the grand canonical partition function includes an additional term for the chemical potential of the system.

5. Can the canonical partition function be used for interacting particles?

No, the canonical partition function is only applicable to non-interacting particles. For systems with interacting particles, more complex partition functions must be used, such as the configurational partition function or the molecular partition function. These account for the interactions between particles and are essential for accurately describing the thermodynamic properties of such systems.

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